Blowing Mathematical Bubbles with Kirchhoff-Plateau

Credit: Okinawa Institute of Science and Technology

Often used for children’s enjoyment and artistic performances, soap bubbles are delicate, light-reflecting films that typically last just a few seconds before bursting. But beyond their value to entertain, soap bubbles are physical examples of the rich mathematical problem of minimal surfaces; they assume the shape of the least surface area possible, containing a given volume. Researchers at the Okinawa Institute of Science and Technology Graduate University (OIST) have recently worked out the solution to a mathematical problem — known as the Kirchhoff-Plateau problem — that is simply illustrated by soap films that span flexible loops.

“Our solution of the Kirchhoff–Plateau problem brings beautiful mathematical results close to what happens in the physical world,” says Dr. Giulio Giusteri, coauthor of the paper which was recently published in the Journal of Nonlinear Science. Dr. Giusteri worked with Professor Eliot Fried, who heads OIST’s Mathematical Soft Matter Unit, and Dr. Luca Lussardi from the Università Cattolica del Sacro Cuore in Italy.

The question answered by the team is a variant of the “Plateau problem”, a centuries-old mathematical problem, named after 19th century Belgian physicist, Joseph Plateau. Plateau hypothesized that when you dip a rigid wire frame into a soap solution, the surface of the soap film formed on the frame represents a minimum mathematically possible area, no matter the shape of the frame.

The first satisfactory solution for the Plateau problem was provided in the 20th century, by American mathematician Jesse Douglas, for which he was awarded the Fields Medal in 1936. More recently, in 2015, Professor Jenny Harrison from UC Berkeley and Harrison Pugh of Stony Brook University extended Douglas’s work, providing a proof valid under general hypotheses encompassing, for example, situations in which junctions are present where multiple soap films meet each other. … (Okinawa Institute of Science and Technology)